| L(s) = 1 | + 6·3-s − 4·5-s − 20·7-s + 27·9-s − 60·11-s − 26·13-s − 24·15-s − 84·17-s − 60·19-s − 120·21-s − 72·23-s − 210·25-s + 108·27-s − 124·29-s − 108·31-s − 360·33-s + 80·35-s + 36·37-s − 156·39-s − 52·41-s − 32·43-s − 108·45-s − 428·47-s − 134·49-s − 504·51-s + 380·53-s + 240·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.357·5-s − 1.07·7-s + 9-s − 1.64·11-s − 0.554·13-s − 0.413·15-s − 1.19·17-s − 0.724·19-s − 1.24·21-s − 0.652·23-s − 1.67·25-s + 0.769·27-s − 0.794·29-s − 0.625·31-s − 1.89·33-s + 0.386·35-s + 0.159·37-s − 0.640·39-s − 0.198·41-s − 0.113·43-s − 0.357·45-s − 1.32·47-s − 0.390·49-s − 1.38·51-s + 0.984·53-s + 0.588·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 4 T + 226 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 20 T + 534 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 60 T + 3114 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 84 T + 10582 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 60 T + 6526 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 72 T + 14430 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 124 T - 1586 T^{2} + 124 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 108 T - 16154 T^{2} + 108 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 36 T + 42382 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 52 T + 76666 T^{2} + 52 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 32 T + 150198 T^{2} + 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 428 T + 116242 T^{2} + 428 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 380 T + 258142 T^{2} - 380 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 1420 T + 882490 T^{2} + 1420 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 1012 T + 708206 T^{2} - 1012 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 844 T + 778238 T^{2} + 844 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 868 T + 888050 T^{2} + 868 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 60 T + 114886 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 272 T + 993374 T^{2} - 272 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1252 T + 1118698 T^{2} + 1252 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 572 T + 853306 T^{2} - 572 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 708 T + 1762390 T^{2} - 708 p^{3} T^{3} + p^{6} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.95022448435788984664116694516, −10.32562671646902159028816148497, −9.993498419882439564928277611599, −9.734403784483738748995039249710, −8.911621503580280151647705856309, −8.842982570830159360008184520910, −8.028751454148506341239027958314, −7.72187164675514945245562756759, −7.34869108573832431349859057970, −6.73373998475477755736772247052, −6.05350067582724653057719832987, −5.64994998988630759394864214797, −4.55562015057282169677214473534, −4.45692626904635966156228758992, −3.33770158113213474685252645889, −3.26800874539872326447758293689, −2.13155032319669596782182198073, −2.06605078619900576989463972130, 0, 0,
2.06605078619900576989463972130, 2.13155032319669596782182198073, 3.26800874539872326447758293689, 3.33770158113213474685252645889, 4.45692626904635966156228758992, 4.55562015057282169677214473534, 5.64994998988630759394864214797, 6.05350067582724653057719832987, 6.73373998475477755736772247052, 7.34869108573832431349859057970, 7.72187164675514945245562756759, 8.028751454148506341239027958314, 8.842982570830159360008184520910, 8.911621503580280151647705856309, 9.734403784483738748995039249710, 9.993498419882439564928277611599, 10.32562671646902159028816148497, 10.95022448435788984664116694516
